Time-Frequency Representations on Lorentz Spaces

Abstract

In this paper we study boundedness properties of the short-time Fourier transform (STFT) on the Lorentz spaces Lp,q(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{p,q}(\mathbb {R}<^>d)$$\end{document}. The same results apply to the cases of the Wigner and tau\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}-Wigner transforms. Reinterpreting these results in terms of operators we obtain boundedness properties for Weyl and tau\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}-Weyl operators. We conclude with an application to the uncertainty principle of Donoho-Stark in the context of Lorentz spaces.